| L(s) = 1 | + 1.43·3-s − 5-s + 1.96·7-s − 0.951·9-s − 5.12·11-s + 1.62·13-s − 1.43·15-s + 5.16·17-s + 6.46·19-s + 2.81·21-s − 0.862·23-s + 25-s − 5.65·27-s + 1.93·29-s + 7.62·31-s − 7.32·33-s − 1.96·35-s + 37-s + 2.32·39-s − 10.1·41-s + 9.83·43-s + 0.951·45-s − 8.66·47-s − 3.13·49-s + 7.39·51-s − 2.89·53-s + 5.12·55-s + ⋯ |
| L(s) = 1 | + 0.826·3-s − 0.447·5-s + 0.743·7-s − 0.317·9-s − 1.54·11-s + 0.451·13-s − 0.369·15-s + 1.25·17-s + 1.48·19-s + 0.614·21-s − 0.179·23-s + 0.200·25-s − 1.08·27-s + 0.358·29-s + 1.37·31-s − 1.27·33-s − 0.332·35-s + 0.164·37-s + 0.372·39-s − 1.59·41-s + 1.49·43-s + 0.141·45-s − 1.26·47-s − 0.447·49-s + 1.03·51-s − 0.398·53-s + 0.690·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.341768685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.341768685\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 1.43T + 3T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 - 1.62T + 13T^{2} \) |
| 17 | \( 1 - 5.16T + 17T^{2} \) |
| 19 | \( 1 - 6.46T + 19T^{2} \) |
| 23 | \( 1 + 0.862T + 23T^{2} \) |
| 29 | \( 1 - 1.93T + 29T^{2} \) |
| 31 | \( 1 - 7.62T + 31T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 9.83T + 43T^{2} \) |
| 47 | \( 1 + 8.66T + 47T^{2} \) |
| 53 | \( 1 + 2.89T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 7.24T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 - 9.03T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 7.24T + 83T^{2} \) |
| 89 | \( 1 - 7.85T + 89T^{2} \) |
| 97 | \( 1 - 2.24T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.356920225959347820566636375220, −8.046824272651225972741032048296, −7.70150137219761216626505248149, −6.58107594974817818514397803116, −5.29959176035515907597534742653, −5.14531779902586907859219822815, −3.75135088626877643288320754527, −3.09727284414516070301063390190, −2.28741462417006036445843477260, −0.916375578361149214345524148066,
0.916375578361149214345524148066, 2.28741462417006036445843477260, 3.09727284414516070301063390190, 3.75135088626877643288320754527, 5.14531779902586907859219822815, 5.29959176035515907597534742653, 6.58107594974817818514397803116, 7.70150137219761216626505248149, 8.046824272651225972741032048296, 8.356920225959347820566636375220
